higher geometry / derived geometry
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Given any object $X$ in a cartesian monoidal category where formal completion is defined, then the formal neighbourhood of the diagonal of $X$ is the formal completion of the diagonal map $\Delta_X \colon X \longrightarrow X \times X$.
Hence intuitively, the formal neighbourhood of the diagonal of $X$ is the space whose points are pairs of infinitesimally close points in $X$.
Specifically for $X$ a scheme, then the formal neighbourhood of its diagonal is the formal scheme around $\Delta_X$. In this case the coequalizer of the two projections out of the formal neighbourhood is called the de Rham stack of $X$. Moreover, the (sheaf of modules of) sections of the formal neighbourhood, with respect to one of the two projection maps, is the tangent complex of $X$. This holds indeed more generally, at least also for $X$ a derived algebraic stack (Hennion 13).
More generally, see at infinitesimal disk bundle.
Last revised on January 9, 2016 at 22:38:39. See the history of this page for a list of all contributions to it.